Stochastic models The former one is used to A random function Z Z
a summary of findings and suggestions for future enhancements in Section 4.
2. MODELS AND VARIOGRAMS 2.1 Random function and variogram
From spatial statistics point of view, the spatial structure and variability revealed in SAR data can be modeled by a
regionalized variable zx, which is simply considered as the realization of a random function Zx constructed at all points x
of a given region D ⊂ R
2
, while Z={Zx : x ∈ D ⊂ R
2
} defines a random field on D.
The SAR data have been translated into general datum for analyzing. A stationarity condition should be set up first. Let
Zx be the second-order stationarity random function, then it is of the following properties,
E Zx
⎡⎣ ⎤⎦= m
Cov Z x,Zx + h
⎡⎣ ⎤⎦= Ch
1
where m is the expectation value of Zx and Ch is the covariance function for all pairs of pixels x and x+h. When h =
0,
Cov Z x ,
Z x + h
⎡ ⎣
⎤ ⎦
= Cov Z x
,
Z x
⎡ ⎣
⎤ ⎦
= Var Z x
⎡ ⎣
⎤ ⎦ =
C 0
2
he covariance function divided by the variance is called the correlation function
C
C h h
=
ρ
3 Under second-order stationarity assumption, the second-order
variogram
γ
2
h of Zx, which describes the variability between the intensity values of two pixels separated with h, is
γ
2
h =
1 2
Var Z x + h − Zx
⎡⎣ ⎤⎦
4 From Eqs. 1 and 2 Schabenberger Gotway, 2005,
2
1 [
] 2
1 [
] 2
1 [ ]
2 [
, ] [ ]
Var Z Z
Var Z Var Z
Cov Z Z
Var Z C
C C
γ
= +
− =
+ +
− +
= −
= −
h x h
x x h
x x h
x x
h h
5
In isotropic stochastic processes, the covariance is a function
of the Euclidian distance h=||h|| only, that is, Ch = Ch.
σ
2
is
the variance of Zx. As a result, Eq. 5 can be simplified as follows,
2 2
h C
h −
=
σ γ
6 Fig.1 shows the relationship between Ch and
γ
2
h. R is the largest distance between two relative positions; Sill is the
limitation value of
γ
2
h, its equal to C0. When h exceeds the range, Ch tends to zero and
γ
2
h is stabilized. It means that data separated by a distance larger than the R are uncorrelated.
The R is an important parameter related to the spatial scale of the data Feng, 2008. It can be used to reflect the floe size of
sea ice, so how to estimate R from data images accurately is a significant part of this method.
For a given random function Zx, under second-order
stationarity assumption and in isotropic stochastic processes, its
first-order variogram
γ
1
h is defined as follow Chiles Delfiner, 1999
γ
1
h = 1
2
E Z x + h − Zx
⎡ ⎣
⎤ ⎦
7 where
γ
1
h is seen as first-order moment. When h approaches its range R, the value of
γ
1
h tends to be constant. The first- and second-order variograms will both be used to characterize the
spatial structures and their validities and accuracies will be compared in the following sections.